Math Questions in Lorentzgroup: Covariance/Contravariance Explained

[Dreak]

Hi, I'm having some difficulties with my course relativity, more specifically the maths part:

To begin with, what is the 'physic' difference between covariance/contravariance four-vector (if they call it like that in English?).

I read that they transform the components of the Vectors, but not the Vectors themselves. Does that mean you got the same vector, but only it's components in an other basis, something like that?

How do you use them in formula (like: are there different math rules to use, or are only the components different between a covariant vector and its contravariant?)

The minkowski metric is same in contra-covariance, but what about other metrics, how do I change them from one to another, with the Minkowski metric or.?
Or do we simply have to apply: g_μvg^vλ = δ(μλ) ?In my course, there is written that a transformation of an element in the Lorentzgroup can be written as:
V^'μ = \Lambda ^μ_vV^v

What are these elements of the Lorentzgroup? What does \Lambda stand for?

Last but not least, what would be the difference between
\Lambda^μ_v and \Lambda_μ^vThanks in advance!

 

[WannabeNewton]

The difference between covariant and contravariant vectors has been discussed multiple times on the forum. Try doing a forum search using the search function at the top right. For example: https://www.physicsforums.com/showthread.php?t=689904

If you are in Minkowski space-time then the isomorphism between the contravariant and covariant indices of any tensor is given by the Minkowski metric.

Usually the subgroup of interest is the proper Lorentz group $SO(1,3)$ but regardless you can find everything you need here: http://en.wikipedia.org/wiki/Lorentz_group

$\Lambda$ usually stands for a Lorentz boost:http://en.wikipedia.org/wiki/Lorentz_boost#boost

In such a case, the matrix representation $\Lambda^{\mu}_{\nu}$ is symmetric so there is no difference between the two matrices you wrote down.

 

[Mentz114]

In my course, there is written that a transformation of an element in the Lorentz group can be written as:
V^'μ = \Lambda ^μ_vV^v

What are these elements of the Lorentzgroup? What does \Lambda stand for?

Last but not least, what would be the difference between
\Lambda^μ_v and \Lambda_μ^v

The Lorentz transformation can be written like this $v^{\mu'}=\Lambda^{\mu'}_\mu v^\mu$

So $v^\mu=\Lambda_{\mu'}^\mu v^{\mu'}=\Lambda_{\mu'}^\mu\Lambda^{\mu'}_\mu v^\mu$ from which you can get the relationship between $\Lambda_{\mu'}^\mu$ and $\Lambda^{\mu'}_\mu$.

I'm not sure if this is what you are asking, though.

 

[Dreak]

Thanks for the help!

The Lorentz transformation can be written like this $v^{\mu'}=\Lambda^{\mu'}_\mu v^\mu$

So $v^\mu=\Lambda_{\mu'}^\mu v^{\mu'}=\Lambda_{\mu'}^\mu\Lambda^{\mu'}_\mu v^\mu$ from which you can get the relationship between $\Lambda_{\mu'}^\mu$ and $\Lambda^{\mu'}_\mu$.

I'm not sure if this is what you are asking, though.

Not completely, I wonder what the difference is between: first the lower indic and then the upper indic or the other way around, first upper and then the lower indic (I hope I'm clear enough O:) )

 

[dextercioby]

[...]Last but not least, what would be the difference between
\Lambda^μ_v and \Lambda_μ^v
[...]

One is the transpose of the other.